Difference between revisions of "Chain"
From Encyclopedia of Mathematics
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| − | A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group | + | ===In ordered sets=== |
| + | The same as a [[totally ordered set]]: in a general [[partially ordered set]], a subset which is totally ordered with respect to the induced order. The [[rank of a partially ordered set]] is the maximal cardinality of a chain. | ||
| + | |||
| + | ===In algebraic topology=== | ||
| + | A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) {{ZBL|0047.41402}}</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) {{ZBL|0091.36306}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 15:33, 18 March 2023
In ordered sets
The same as a totally ordered set: in a general partially ordered set, a subset which is totally ordered with respect to the induced order. The rank of a partially ordered set is the maximal cardinality of a chain.
In algebraic topology
A formal linear combination of simplices (of a triangulation, of a simplicial set and, in particular, of singular simplices of a topological space) or of cells. In the most general sense it is an element of the group of chains of an arbitrary (as a rule, free) chain complex. A chain with coefficients in a group $G$ is an element of the tensor product of a chain complex by the group $G$.
References
| [1] | N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966) Zbl 0047.41402 |
| [2] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) Zbl 0091.36306 |
How to Cite This Entry:
Chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain&oldid=12098
Chain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chain&oldid=12098
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article